analysis of optimization of blank holding force in deep drawing … · 2018-03-09 · stamping and...

Gyadari Ramesh, Dr.G.Chandra Mohan Reddy / International Journal Of Engineering

Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com

Vol. 3, Issue 4, Jul-Aug 2013, pp.1975-1995

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Analysis of Optimization of Blank Holding Force In Deep

Drawing By Using LS DYNA

Gyadari Ramesh, Dr.G.Chandra Mohan Reddy Department of Mechanical Engineering, Osmania University, Hyderabad-500085.

ABSTRACT Sheet metal forming problems are typical

in nature since they involve geometry, boundary

and material non-linearity. Cup drawings involves

many parameters like punch and die radius,

clearance, lubrication, blank holding force and its

trajectories etc. So designing the tools for cup

drawing involves a lot of trial and error

procedure. To reduce number of costly trial error

steps, the process can be simulated by using finite

element packages. Even the finite element package

gives an approximation towards the solution. The

experimentation is inevitable. The aim is to study

analysis of optimization of blank holding force

developed for cup drawing operation by using

explicit finite element package LS DYNA. One of

the basic problems in deep drawing is wrinkling.

Wrinkling can be avoided by using blank holding

force. But higher the blank holding force (BHF),

higher is the frictional force, so more will be

tensile stresses in cup wall there by promote

tearing failure at the punch corner. Hence BHF

needs to optimized so as to prevent the wrinkling

and at the same time to prevent tearing failure. In

this work die design is done for a cup of 30mm

diameter and deep with 1 mm thickness. For the

same blank holding force is calculated from the

empirical formula. The same is simulated on an

explicit finite element package LS DYNA. By an

iterative procedure the optimum blank holding

force is obtained and presented. B H F in deep

drawing is an essential parameter to be

determined optimally to avoid formation of

wrinkles. It is also necessary to at the same time to

determine the force in drawing operation and

failure of the cup. Higher the B H F, higher is the

frictional forces between the blank and blank

holder, so higher the loads required for drawing

operation and higher the strains developed in the

cup walls between the die and punch, thereby

reducing thickness of the section. In this thesis

optimum blank holding force has been found out

by checking the condition of nonformation of

wrinkles at different coefficient of friction at

(0.045, 0.06, 0.1, 0.13, and 0.15) and at different

die radius (2, 3, 4, 5,6mm) and the values of blank

holding force has been taken where no wrinkles

has been formed for different coefficient of

friction and for different die radius and the

graphs are plotted and the results are studied. h-

Method is used for mesh convergence stability of

max vonmises stress is taken as a parameter to

check the convergence.

Keywords – Deep Drawing by Using LS DYNA,

Blank Holding and blank holding force (BHF).

I. INTRODUCTION Sheet metal forming is one of the most

widely used manufacturing processes for the

fabrication of a wide range of products in many

industries. The reason behind sheet metal forming

gaining a lot of attention in modern technology is due

to the ease with which metal may be formed into

useful shapes by plastic deformation processes in

which the volume and mass of the metal are

conserved and metal is displaced from one location to

another. Deep drawing is one of the extensively used

sheet metal forming processes in the industries to

have mass production of cup shaped components in a

very short time. In deep drawing, a flat blank of sheet

metal is shaped by the action of a punch forcing the

metal into a die cavity Sheet metal forming is one of

the most common manufacturing processes to

plastically deform a material into a desired shape.

Products include hundreds of automotive

components, beverage cans, consumer appliances,

submarine hulls, and air craft frames. Based on the

geometry, the volume and the material, sheet metal

forming can be divided into various categories such

as stamping, deep drawing, stretch forming, rubber

forming, and super plastic forming. Among these,

Stamping and deep drawing are the most common

operations.

Deep drawing products in modern industries

usually have a complicated shape, so these have to

undergo several successive operations to obtain a

final desired shape. Trimming of the flange is one of

those operations and that is used to remove the ears

i.e. to have uniform shape of the flange on all the

sides of the final product. These are formed due to

uneven metal flow in different directions, which is

primarily due to the presence of the planar anisotropy

in the sheet.

The main concern of the deep drawing

industry is to optimize the process parameters in

order to get a complete deep drawn product with least

effects and high limiting drawing ratio. In order to

achieve this optimization objective a large number of

solution runs need to be performed in order to search

for the optimum solution. Furthermore, the quality of




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the products can be increased. With reference to an

economical success it is very important to put better

and cheaper products faster on the market than other

competitor‟s. A substantial aid for this is the

numerical simulation. Costs and time for tool

adapting could play an outstanding roll. Furthermore,

changes in design while fabricating a prototype are

usual. By means of numerical simulation, potential

forming problems can be recognized during

fabricating a first tool. Despite many advantages of

the numerical simulation, it must be said, that there

are costs for hardware, software, training and for the

simulation itself. However, it is an effective means

for making forming processes and new products

cheaper. Tool loads can be computed and overloads

can be predicted by means of FEM, which is very

difficult in practical experiments.

The depth of draw may be hallow, moderate

or deep. If the depth of the formed cup is more than

its diameter, the process is called Deep Drawing.

Parts of various geometric and sizes are made by

drawing operation, two extreme example being bottle

caps, automobiles panels etc. the simplest example is

the drawing of a flat bottom cylindrical cup.

In the drawing of a cylindrical cup, a round

sheet metal blank, is placed over a circular die

opening and is held in place with a blank holder. The

punch travels downward and forces the blank into the

die cavity, forming a cup. The important variables in

deep drawing are the properties of sheet metal, the

ratio of blank diameter to punch diameter, the

clearance between the punch and die, the punch

corner radius and die corner radius, the blank holder

force, friction and lubrication. The forces occurring

during drawing are bending at the radii, friction

between blank holder and sheet metal, die and sheet

metal, punch and sheet metal and compression at

flange area or extremity of cup. Usually Drawing is a

process of forming a flat, pre-cut, metal blank into a

hollow shape, either cylindrical or box-shaped, by

pressing it into a die cavity without excessive

wrinkling, thinning, or fracturing. Typical parts

produced by drawing include beverage cans,

containers of all shapes and sizes, and automobile

and aircraft panels. Deep drawing process is

influenced by some parameters like residual stresses,

Blank holding force etc.

Residual stresses also play a very important

role in how a formed part in a deep drawn cup. These

stresses can become so large in a deep drawn cup that

cracks are formed in the cup wall. These residual

stresses can be removed by annealing the cup right

after the deep drawing. However in most cases it is

desire to avoid the annealing process. This process

increases the production costs, and can lead to an

inexpedient production flow and can give problems

with regard to maintaining close tolerances due to

distortion during annealing process.

B H F is an important parameter in deep

drawing process. It is used to suppress the formation

of wrinkles that can appear n the flange of the drawn

part. When increasing the B H F, stress normal to the

thickness increases which restrains any formation of

wrinkles. However, the large value of

the B H F will cause fracture at the cup wall and

punch profile. So, the B H F must be set to a value

that avoids both process limits of wrinkling and

fracture. Avoid wrinkling and tearing such that at ach

punch travel (L), the following relations must be

satisfied:

FBH >F wrinkling and FBH <F Tearing

A given technical problem must be expressed by

physical terms so that it can be formulated

mathematically, what means modeling. The model

should reflect the reality as exactly as possible.

However, it should also be as simple as possible.

Furthermore, the model must be described this way

that it can be implemented in computers. Numerical

problems like divisions by extremely low numbers

or poor convergences of iterations, respectively,

have to be mastered or to be avoided. Trial runs of

the computational simulations and a subsequent

check of the results by comparison with reality or

physical experiments are a must. A special attention

has to be directed to the boundary and initial

conditions during modeling because they have a

decisive influence on the extent of the model as well

as on its reliability. If the results do not coincide

with reality or with the expectations close to reality,

the model must be checked and possibly modified,

whereby it will become bigger and more

complicated.

II. HEADINGS

I.INTRODUCTION

II. LITERATURE SURVEY

2.1. Sheet Metal Forming

2.1.1. Objective

2.1.2. Types of Forming

2.1.3. Theory of sheet metal drawing

operation.

2.1.4. Reducing severity of draw

2.2. Deep Drawing

2.3. Wrinkling & Tearing

2.4. Blank Holding Force

III. MATHEMATICAL FORMULATION

3.1. Co-rotational coordinates

3.2. Velocity-Strain Displacement Relations

3.3. Stress Resultants and Nodal Forces

3.4. Material properties

3.4.1. Power Law Isotropic Plasticity

3.4.2. Rigid Material

IV. Design Calculations

4.1. Blank diameter

4.2. Allowances for Trimming




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4.3. Percentage reduction

4.4. Thickness to diameter ratio

4.5. Drawing Force

V. FINITE ELEMENT SIMULATIONS

5.1. Selection of optimum mesh size for the

blank

5.2. Selecting optimum Blank Holding Force at

different Coefficient of Friction

5.3. Selecting optimum Blank Holding Force at

different Die Radius

5.4. Variations of vonmises stresses with blank

holding force at different Coefficient of

Friction

5.5. Variation of plastic strain with Blank

Holding Force at different Coefficient of

Friction

5.6. Variation of Vonmises stresses with Blank

Holding Force at different Die Radius

5.7. Variation of Plastic strain with Blank

Holding Force at different Die Radius

VI.SIMULATION RESULTS

VI.CONCLUSION

6.1 Conclusion

REFERENCES

II. OBJECTIVE

The main objective of the proposed research

is to find the optimal blank holding force that is to be

used in the deep drawing process to produce a cup of

required shape and size without wrinkles at different

coefficient of friction and at different die radius. For

general deep drawing operations, most people define

the optimal blank shape as that blank profile which

can be deformed into a cup with either a uniform

flange profile or uniform rim height i.e. cup free from

ears. However it is not easy to find an optimal blank

holding force because of complexity of deformation

behavior and there are couple of process parameters

like die radius, punch radius, punch speed, blank

holder force and amount of friction which affects the

result of the process i.e. tearing, wrinkling, spring

back and surface conditions such as earing. Even a

slight variation in one of these parameters can result

in defects. Until now, the optimal blank holding force

along with the input of optimal process parameters is

performed by a trial and error method based on the

expertise of the engineer. But recently, in order to

address the change of demand from mass production

to batch production for higher quality products in

ever shorter time, this experimental trial and error

technique has turned out to be very expensive and

time consuming. Therefore, numerical simulations of

sheet metal forming processes based on the finite

element method (FEM) represent a powerful tool for

prediction of forming processes.

III. LITERATURE SURVEY: 3.1 Sheet Metal Forming:

In metal forming, a piece of material is

plastically deformed between tools to obtain the

desired product. A special class of metal forming

concerns the case where the thickness of the piece of

material is small compared to the other dimensions,

i.e. sheet metal forming. Sheet metal forming is a

widely used production process: in 1998, 265 million

tons of steel sheet and 9 million tons of aluminum

sheet was produced worldwide which was

approximately 35% of the total steel and aluminum

production [Langerak, 1999a][1]. Sheet metal

forming is characterized by a stress state in which the

component normal to the sheet plane is generally

much smaller than the stresses in the sheet plane. A

commonly used sheet metal forming process is the

deep drawing process. The principle of deep drawing

is schematically represented in Figure 3.1.

Fig. 3.1 Schematic of deep drawing process

Haar and carleer [2] mentioned the material

flow into the die cavity is controlled by the blank

holder; a restraining force is created by friction

between the tools and the blank. The friction between

the tools and the blank is influenced by the blank

holder force, lubrication or by coatings on the blank

or tools. The work as described is this thesis is

implemented in the implicit finite element code

DiekA. The finite element code DiekA developed at

the University of Twente, is a multi-purpose package

which is able to simulate various forming processes

such as rolling, deep drawing, extrusion, cutting and

slitting. The deep drawing part of this code was

developed in close cooperation with Hoogovens

Research and Development, a part of the Corus

Group PLC since October 6, 1999. The development

of the deep drawing part of DiekA was started in

1987. In 1992, Vreede[3] presented deep drawing

simulation results of axi-symmetric products,




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rectangular products and a simple automotive

product, making use of a 3-node triangular element

based on membrane theory (i.e. only incorporating

stretching energy) [Vreede, 1992]. The material

behavior was described by rigid plastic constitutive

relations and the planar isotropic Hill yield criterion.

The contact behavior was described by special

contact elements and Coulomb friction. Finally, the

tools were numerically described by a collection of

measurement points or by elements. In 1992, the

work of Vreede was continued by Carleer. The new

developments were focused on improving the

existing code in order to better satisfy the

requirements for industrial application. In the

subsequent five years, the following improvements

were implemented [Carleer, 1997]: two new 3-node

triangular element types, i.e. an element based on

Kirchhoff theory (incorporates membrane and

bending stresses) and an element based on Mindlin

theory (incorporates membrane, bending and shear

stresses). The anisotropic behavior of the material

was taken into account by implementing the

anisotropic Hill‟48 yield criterion and the Vegter[4]

yield criterion based on multiaxial stress states

[Vegter, 1999]. An elastoplastic constitutive relation

was implemented in order to predict the springback

behavior after deep drawing. The contact description

was improved by a fast contact search algorithm and

a more sophisticated friction model. Finally, an

equivalent draw bead model was developed to

efficiently incorporate draw beads in a finite element

simulation. Sheet metal forming is characterized by

large relative displacements between the sheet and

the tooling, spatial and temporal strain variations in

the part and complex boundary conditions. Robust

and accurate analysis is needed to simulate forming

process and defects, with the ultimate goal to

eliminate costly die-tryouts, particularly when

introducing new materials and processes. Finite

element models must be computationally efficient for

practical use with today‟s computers and those in

foreseeable future. Wang and budiansky[5]

introduced the membrane element formulation based

on a theory of shells presented by budiansky[6].

When compared to large strain, large displacement,

and elastoplastic shell formulations {7-9}, membrane

formulations have been reported to be 5 times faster

[10-12]. Earlier research in our group showed of 5x

to 20x for shell simulations versus membrane ones.

3-D continuums are seldom applied to general

forming because of limited computation time [10-11].

The computational efficiency of membrane elements,

make them attractive for arbitrary 3D geometries,

but they fail to produce a convergent solution for

bending-dominated forming problems, or to

reproduce bending effects such as flange/ wall

wrinkling or spring back.

Hybrid methods based on empirical results

introducing bending effects into membrane sheet

forming programs have been proposed. Stoughton

[10] reduced the terms of tangent stiffness matrix

using a draw bead model of Wang [14]. Good results

were reported for R/t ratios ranging from 3.1 to 18.4

with minimal increase in computer time,

approximately 6%. Pourboghart and chandorkar [15]

used the contact conditions, stress and strain states

and curvatures of the tooling to adjust the membrane

solution (after computation), and to calculate spring

back and side wall curl. Although both approaches

were computationally efficient, they have proven

difficult to generalize to arbitrary geometries. For

linear-elastic problems, corrections to the membrane

residual vector and corresponding stiffness matrix

have been formulated [16-119]. Inter element

bending was evaluated from the relative rotations of

adjacent two elements; the bending stiffness is being

represented by torsional springs of a specified

stiffness. Similar approaches have been

developed for non linear sheet forming problems,

where the sheet is discretized into two superimposed

meshes accounting for bending and stretching [20-

26]. Keum [20] introduced this concept without the

mechanical formulation for FE implementation. Huh

et al. [21-23] derived a family of „Bending Energy

Augmented Membrane‟ (BEAM) elements with

rotational springs at nodes or at element edges. The

bending stiffness of this matrix for these elements

were assumed as constant during a time step and

updated at the end of time step. Six-noded patches of

four constants strain triangular elements have been

used to compute the inter-element bending forces for

dynamic explicit programs based on an elastic

constitutive law [24] and elastic plastic constitutive

law [25].

3.1.1 TYPES OF FORMING

Many forming operations are complex, but

all consists of combinations or sequences of the basic

forming operations bending, stretching and drawing.

3.1.1.1 Bending:

Bending is the metal working process by which a

straight length is transformed into a curved length it

is a very common forming process for changing sheet

and plate into channels, drums tanks etc. during the

bending operations, the outer surface of the material

is in tension and the inside surface in compression.

The strain in the bent material is increases with

decreasing radius of curvature. The stretching of the

bend causes the neutral axis of the section to move

towards the inner surface. In most cases, the distance

of the neutral axis from the inside of the bend is 0.3t

to 0.5t, where “t” is the thickness of the part.

3.1.1.2 Stretch forming:

Stretch forming is the process of forming by

the application of primarily tensile forces in such a

way as to stretch the material over a tool or form

block. Stretching is caused by tensile stresses in




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excess of the yield stress. When they are applied in

perpendicular directions in the plane of the sheet,

these courses produce biaxial stretching. When the

perpendicular forces are equally balanced biaxial

stretching occurs. Much higher levels of deformation

as measured by an increase in area can be reached in

balanced biaxial stretching than in any other forming

mode. Many forming operations involve stretching of

some means within the stamping. Automotive outer

body panels are typical examples of parts formed

primarily by stretching. Parts with regions containing

domes (microwave reflectors), ribs and embossments

also involve stretching.

3.1.2 Theory of sheet metal drawing operation

Many irregular shaped parts are drawn, and

the theories of metal flow in these parts are

complicated. The drawing of cups is the simplest

drawing operation and more easily illustrated.

Therefore the remaining discussion refers to the

operation known as cupping. The blank required for

cupping is round. An analysis should be made of

what happens as the punch and die first starts to draw

the blank. The blank edge is forced down to a smaller

circumference; such a reduction means that a

compressive force is being applied to the metal.

3.1.2.1 Metal flow:

Drawing operation consists of metal flow

rather than metal movement. These terms were

described in the theory of forming sheet metal.

During cupping, the metal flows into the cup shape,

the metal follows itself into the cup shape. There is

no movement of metal through space as there is

during a forming operation, the metal flows through

the opening provided by the clearance between the

punch and die and blank holder. Since the punch

exerts the force on the cup bottom to cause the

drawing action, considerable stretching of metal

occurs in the cup side wall near the cup bottom. Fig

3.2

Fig 3.2 Forces during Cupping

Fig 3.3 Metal flow During Cupping

shows metal flow in cupping die. Figure3.3 shows

the forces involved on the outer edge of the blank,

this metal tends to thicken. The thinning and

thickening of metal in the cupping operation also

may be referred to as metal flow.

1.2.2 Forces during drawing:

The forces occurring during drawing are:

1. Bending at the radius

2. Friction between

a. Blank holder and sheet metal

b. Die and sheet metal

c. Punch and sheet metal

3. Compression at flange area or extremity of

cup.

The punch exerts a force on the cup bottom

of sufficient magnitude to overcome the sliding and

stationary friction, to bend force it at the radii and to

compress the metal at the cup extremity. Therefore

the punch force is the sum of the other three. The

punch force is the applied force and the other three

are totaled to obtain the equal and opposite reaction

force.

Wrinkles are formed due to improper design

of the die. The die has to given a taper in order to

increasing the load force. This may also happen due

to the improper lubrication. The force is transmitted

by the cup side wall. The wall is placed under tension

at a point near the cup bottom. If the punch force or

the total of the other three forces exceeds the ultimate

tensile strength of the metal, the wall will break. The

final analysis of force is as follows:




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Friction + compression + bending =

punch force

The punch force must be less than the ultimate

strength of the metal or failure will occur.

3.1.3 Characteristic cupping force curve:

The curve representing the force required

during cupping reveals many characteristics of

drawing. The maximum force occurs at the first

instant of cupping. Therefore, cupping consists of

high instantaneous force, which immediately reduces

after metal flow has been started. The base of the

cupping force curve represents the depth of the cup.

The graph shown in figure 3.4 between punch force

and punch stroke for drawing operation gives the

total punch force, friction, ideal deformation and

ironing.

Fig 3.4 Punch force vs. punch stroke for drawing

operation

The main conclusion that can be made from

the curve is that the initial contact of the punch steel

with the metal blank is the point of severest stress or

strain. If wrinkles occur, they start while this high

force is applied. If the cup breaks, failure occurs

while this high force is applied. Once metal flow has

started and the force is reduced, the chances of

wrinkles or cup breaking are limited. If cup failure

occurs after this point, the failure can usually be

attributed to inclusions or defects in the sheet metal.

As long as blank size and punch diameter are being

used. Whether or not a flange is left on the cup has no

effect on the severity of the operation. This condition

is illustrated in figure 3.5. The measure of the

severity of drawn is found by comparing the punch

diameter. For irregular shaped drawing, a comparison

of the blank area will the initial contact area of the

punch would be an indicator of

severity.

Fig 3.5 Severity of drawing

Drawing severity is determined by the relationship

between the punch diameter and blank diameter. The

larger the difference in diameter the greater the

severity in drawing. Greater severity means there is

more tendencies for wrinkling and tearing. When the

punch diameter and blank diameter are constant, the

severity of drawing is not reduced because a flange is

left on the cup. If no flange is left on the cup, the

severity of draw is not appreciably increased. This is

because of small difference in force requirements as

illustrated above.

3.1.4 Reducing severity of draw:

Several methods may be employed to reduce

the severity of draw. Any method that increases the

relative punch contact area will reduce the severity. It

is assumed that a drawing lubricant or compound is

being used to permit free metal flow. The blank

holding pressure must also be correct.

The severity of the draw may be reducing by the

following methods:

1. Increase draw radii

2. Change blank holding surface to an angle

3. Provide lead in or chamfer on die

4. Pre fold the blank with the blank holder

wrap

5. Pre fold the blank before placing it in the

draw die

6. Use redrawing to obtain final size

3.1.4.1 Draw radii:

If possible the radius on the punch is made

the same as the part print radius. The same is true of

the radius on the die. Many times, however, these

radii are too small and increase the severity of the

drawing operation. A small radius restricts the flow




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of metal over the radius. When metal flow is retarded

in such manner, a higher force is required to start the

flow. This higher force is converted to greater tension

on the cup side wall and may cause failure. This

failure may occur before the cup is completely

drawn.

Therefore to reduce the severity of drawing

and reduce metal failure, the radius on the punch and

die is increased. Because more metal flow occurs

over the die radius this is the most critical radius.

3.1.4.2 Redrawing:

When simpler methods fail, the redraw operation is

used to reduce severity. The diameter of the draw

punch is increased to reduce the draw severity. The

difference between the blank and punch diameter is

reducing. The draw punch diameter is increased to a

point where successful draws can be made. This

means that the cup produced is too large in diameter

and too short in height when compared to the part

print. Therefore, secondary operation called

„redraws‟ are necessary to reduce the diameter to part

print specification. One or more draws may require.

Drawing, then, is converting cups from flat blanks.

Redrawing is reducing a cup or drawn part to a

smaller diameter with resulting increase in height.

After drawing, the metal is in a work hardened state.

Therefore redrawing must be consecutively less

severe in reduction to prevent failure of the metal.

Often it is necessary to anneal parts before or

between redrawing operations.

3.1.4.3 Restricting metal flow:

On irregular shaped blanks and panels it is

often desirable to restrict the flow of metal into the

die. Metal flow must be restricted when an access of

metal is flowing into an area. Adding draw beads or

spleens on the blank holding surface restricts flow.

When metal flow must be stopped altogether, the

lock bean may be employed. The bead restricts metal

flow by causing the metal to bend and unbend.

Varying the bead contour and height may alter the

degree of bead restriction. The bead leaves a

depression in the scrap area of the metal, which is

subsequently trimmed off. Decreasing the draw

radius can also restrict metal flow and using a

horizontal blank holding surface can also restricts

metal flow. Serrations in the blank holding surface

will also restrict metal flow. Varying lubricant

consistency on using no lubricant at all may also

control metal flow.

3.1.4.4 Trimming:

Because the blank must be gripped to

prevent wrinkles and because the metal is often

scratched or scored when moving under the blank

holder, excess metal is provided in the blank, this

excess score metal is cut away by the trimming

operation. Bed depressions are also contained in this

scrap.

3.2 Deep drawing

In a deep drawing process, the blank is

deformed into its final desired shape by displacing a

punch into a die and deforming the central region of

the blank. The punch force deforms the blank by

straining it against a constraint which is created by

clamping the blank between a die and a blank holder

along its periphery. The force used to clamp the

blank between the die and blank holder is called the

blank holder force ~BHF!. The blank holder force

can be varied to achieve many desired objectives,

from preventing the occurrence of tearing or

wrinkling, and therefore, increasing the draw depth

[25–27], to controlling springback [28,29]. The blank

holder force can also be varied spatially, by

employing segmented binders, flexible binders and

local adaptive controllers [30–34]. Variation of blank

holder force ~BHF! can be determined a-priori, as

applied in an open-loop manner, or by using a

feedback loop, as a result of feedback control. One of

the first examples of work done on blank holder force

control was that of Hardt and Lee [35]. Using the

general concept of a safe region between wrinkling

and tearing, they proposed two closed-loop control

strategies for a conical cup forming. The first method

was designed to maintain a constant blank holder

displacement, allowing a limited amount of flange

wrinkling. The blank holder force was kept at the

minimum necessary to prevent buckling in the

unsupported region and also to prevent tearing. The

second approach tried to control the binder force by

regulating the volume of material entering the die

cavity, through a generalized thickness parameter

(t*). The conclusion drawn was that the strategies did

not appreciably increase the maximum cup height,

but did significantly reduce the sensitivity of this

maximum value to changes in the blank holder

control variable. Kergen and Jodogne [36,37]

performed studies aimed at determining minimum

BHF curve trajectories for various steels, based on a

wrinkle detection system that measured the distance

between the die and the blank holder in a cylindrical

cup forming. The authors found that the measured

BHF trajectories and the minimum BHF obtained

from experiments varied significantly with variations

in the types and properties of steels being tested.

Hirose et al. [38] showed the success of an increasing

linear combination BHF pattern in preventing the

formation of wrinkles in an automobile panel. The

authors further concluded that if a decreasing, linear

combination BHF trajectory is used, body wrinkles

are not suppressed.

Other researchers have reported favorable

results obtained with decreasing BHF profiles. Kirii

et al. [39], who also tested different linear

combination patterns of BHF in panel formation,

concluded that a decreasing BHF scheme was the

optimum approach. Ahmetoglu et al. [40] obtained a

different set of results. The authors employed




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computer simulation in the drawing of a round cup,

in which three variables ~punch force, radial stress

and thickness strain! were used to control the blank

holder force during simulation. The authors smoothed

the results into a single decreasing BHF trajectory,

which was then used to draw a cylindrical steel cup.

This decreasing trajectory was used to successfully

increase the draw depth over the case of a constant

binder force. Ahmetoglu et al. [41] further examined

decreasing binder force trajectories with regard to the

deep drawing of rectangular parts from aluminum

alloy 2008-T4. Their experiments indicated that a

decreasing binder force significantly reduced the

amplitude of wrinkles, while avoiding the fracture

associated with high BHF values. The work of Sim

and Boyce [42]. The authors performed axisymmetric

cup forming process simulations based on the

tangential force and normalized average thickness

trajectories. These models yielded numerical results

for BHF trajectories that were later employed

to increase the height to which cups could be drawn.

Cao and Boyce [27] built upon this work to develop a

novel approach to determine a variable BHF

trajectory. The authors performed finite element

simulations with PI control of the blank holder force.

They were able to calculate a BHF trajectory having

a combined upward and downward portion that

showed a 16% increase in forming height over the

results obtained by the best constant binder force

case. Recently, experiments by Siegert and Ziegler

[43] have shown that the onset of wrinkling in a

blank drawn with a pulsating BHF occurs at a

displacement similar to that obtained under a constant

BHF equal to the maximum force of pulsation. The

reduction in the friction force achieved due to the

pulse allows more material flow to take place, thus

reducing the chances of tearing. Hsu et al. [44]

proposed an approach for modeling sheet metal

forming for process controller design. They

developed a process model for U-channel forming,

i.e., a mathematical relationship between the blank

holder force and the punch force was determined and

validated experimentally. Characterization of model

uncertainty due to blank size, sheet thickness,

material properties and tooling shape was also

studied. The process model was shown to be effective

in describing the forming process. Blank holder force

variation has also been used to effectively control

spring back in sheet metal forming. Using the

concept of intermediate restraining, Cao et al. [28]

used a neural network to determine a stepped binder

force trajectory that was used to minimize springback

and also obtain consistent results in channel forming,

despite the presence of material variations and

different lubricants. The approach was shown to be

robust and applicable to a wide range of materials

and process conditions. Liu et al. [29] used a similar

approach in the forming of U-shaped parts and

concluded that forming quality was improved when a

variable binder force trajectory was used.

The use of segmented tooling and flexible

binders is an area of sheet metal forming that has also

been gaining prominence in recent years. The

advantages of spatial variation of blank holder force

have been cited in several research endeavors. One of

the first examples of segmented binder tooling can be

found in Siegert et al. [30]. The authors discussed a

deep drawing apparatus developed at the Institute of

Metal Forming Technology in which the lower binder

is composed of eight individual segments, four corner

segments and four straight segments. Each of these

segments is powered by its own separate hydraulic

cylinder. This allows an optimal blank holder force to

be applied to individual regions of the blank. Thus,

the blank holder force can be varied spatially in such

a manner that individual segments of the binder can

apply optimal values of blank holder force as desired.

Neugebauer et al. [31] performed studies using

flexible binders and multiple draw pins. Their

experimental set up consisted of an asymmetric part

and a binder which had 12 draw pins distributed

evenly along its periphery. The draw pins could be

used to apply different values of binder force. They

studied four cases, a rigid binder ~80 mm thick! with

a uniform pin force, a rigid binder with a non-

uniform pin force, a flexible binder ~30 mm thick!

with a uniform pin force and a flexible binder with a

nonuniform pin force. Although no major difference

was observed in the two cases where a rigid binder

was used, the case of a flexible binder with constant

pin force increased the maximum achievable drawing

depth from 70 mm to 90 mm. In the case of a flexible

binder with non-uniform pin force, draw depths up to

110 mm were achieved. Doege et al. [32] proposed

an innovative concept in which the blank holder is

designed as an elastically deformable thin steel plate.

The authors used FEM analysis to determine the plate

thickness and the location of support elements

holding the binder. They performed experiments at

various binder force values to estimate a „„safe

working area.‟‟ The authors were able to show that

the safe working area for a part is larger with a

pliable blank holder and it moves towards higher

blank holder force values. Furthermore, it was shown

that the distribution of pressure on the blank was

more uniform, thus giving rise to improved part

quality. Kinsey et al. [33] proposed a novel method

of forming tailor welded blanks that incorporated a

segmented die with local adaptive controllers. The

local adaptive controllers consisted of hydraulic

cylinders positioned in such a manner as to create an

additional constraint within the forming area. The

position of this constraint is selected so as to

minimize the weld line movement and therefore

reduce the strain developed in the thinner material.

Experiments performed on an asymmetric part




1983 | P a g e

showed that this method of forming helped increase

the draw depth by 20% over the conventional case.

The deep drawing process is applied with

the intention of manufacturing a product with a

desired shape and no failures. The tools, the blank

and the process parameters define deep drawing final

product shape. An incorrect design of the tools and

blank shape or an incorrect choice of material and

process parameters can yield a product with a

deviating shape or with failures.

In the deep drawing the most common defects are

presented in figure 3.6 [45]. In figure 3.6, the

notations are: 1. flange undulation; 2. side

undulation; 3. piece wrinkling; 4. circular traces; 5.

scratches; 6. orange peel; 7. Luders bands; 8.

cracking of piece flange or piece bottom; 9. edges

cracking; 10.- disalignment; 11. Contour

disalignment; 12. festoons; 13. delaminations; 14.

edge festoons [1]. Some of these are the result of the

dies (5, 9, 10, 14), another are due to friction

conditions (1, 4, 13), others are as the result of the

material composition (6, 13) and of the mechanical

properties of the material (1, 2, 3, 6, 7, 8, 11) and

others are as the result of the piece form (12, 14). For

deep drawing by stretching only defects of types 3, 6

and 8 are common [46].

The forming limit curves, FLDs, is one of

the method in examining the failure potential, which

include a good representation of material‟s stretch

ability and the easiness when used for trouble

shooting.

In the design of deep drawing processes,

many process parameters, e.g. die geometry, initial

blank shape, blank holding force, and lubrication,

need to be determined. It is critical to the industry

that these design variables are optimized for one

particular forming process.

The design process has traditionally relied

on experience and intuition accompanied in many

cases by physical trial and error approach. With the

development of computer technology, numerical

simulations using, for example, incremental finite

element methods (FEM) have been developed.

Fig. 3.6. - Types of defects in deep drawing

However, the majority of the applications

using finite element methods has been deterministic,

in which a set of design parameters is given and

simulation results are interpreted by users. Additional

simulations need to be performed if the results are not

satisfactory. The determination of those design

parameters is again mainly based on users' experience

and interpretation. Such design process usually

requires enormous amount of time and cost to

determine the optimal process parameters. Since

metal forming processes are generally highly non-

linear and history dependent, many researchers have

used the direct di_erentiation method for the

calculation of sensitivity [46,52]. Badrinarayanan and

Zabaras [46; 47] calculated the di_erentiation terms

directly from the weak form of equilibrium equations

and demonstrated the approach on determining a

preform shape of an upsetting problem and a die

profile of an extrusion problem. Alternatively, the

di_erential terms can be derived from the nite

element discretized weak form as described in

Chenot and co-workers [47,48]. They calculated a

preform shape of an upsetting problem and tool

shapes of a two-step forging problem. Zhao and

colleagues [49,50] developed a similar optimization

scheme to that of Chenot and co-workers. They

focused on the optimal design of die shapes of

performs rather than that of preform shapes. Kleiber

et al. [51,52] derived sensitivity equations by both the

direct di_erentiation method and the adjoint variable

method. Joun and Hwang [53] used the adjoint

variable method to optimize the die pro_le of a three-

dimensional steady-state extrusion problem. Most

sensitivity analyses for incremental FEM have been




1984 | P a g e

applied to simple two-dimensional problems such as

upsetting, forging, and extrusion. However, there is

very little practical application to sheet metal forming

processesand many di_culties still remain in the

implementation of sensitivity analysis to practical

and complex bulk forming problems.

3.3 Wrinkling & Tearing

Wrinkling is one of the major defects in

sheet metal forming, together with tearing, spring-

back and other geometric and surface defects. Many

research efforts have been dedicated to predict the

occurrence, the location, and the shape of the

wrinkles as evident in the review articles, which

emphasized the work initiated from Hill‟s general

theory of uniqueness and bifurcation. Here the

importances of material models on the prediction

results have been discussed. In most of the finite

element analyses (FEA), accumulated computational

error can be treated as an artificial imperfection in the

system to initiate the wrinkling, but the necessary

magnitude to initiate wrinkling is problem dependent.

This nature determines the uncertainties associated

with the FEA in practical applications. Cao [54]

proposed a stress-based wrinkling predictor. Cao et

al. [54] developed a multi-scale approach with mesh-

free adaptivity to simulate the wrinkling behavior of

sheet metals, while using the stress-based predictor to

generate the desired enrichment function. This

method is now able to capture the wrinkling and post-

buckling behavior in an elastic plate and wedge strip

examples. With all the numerical work, it is

necessary to have well-controlled experiments to

verify the numerical approaches. The experimental

work can be divided into two categories. The first

category includes simple tests of special specimens

designed for simulating the basic mechanism. Sheet

buckling/wrinkling is one of the critical issues not

only in sheet metal forming itself but also in the

numerical simulation of this process. Buckling can

limit the ability to stretch sheet metal during

processing and adversely affect final part appearance

and functionality; severe wrinkling may damage or

even destroy dies. Due to the complexity of buckling,

much research on this topic has been carried out case

by case for a given process by experiment. Yoshida

et al. [55] investigated the material resistance to

wrinkling by stretching a square sheet along one of

its diagonals. A deep drawing of a conical cup and

square cup was analyzed by Senior , Jalkh et al. [54].

Zhang and Yu [56] conducted research on drawing of

elastic–plastic circular plates using a spherical punch.

Besides experimental work, analytical investigation

on the plastic wrinkling initiated from the study on

columns by Shanley [57], followed by the study on

plate and shells by Tvergaard [58] and Needleman ,

V.Tyergaard also prosed a similar concept[59].

General analytical study of plastic wrinkling began

with Hill‟s 1958 bifurcation and uniqueness theory

and was detailed for plates and shell in Hutchinson

(1974)[60].the theory is essentially an Eigen value of

the bifurcation function. Hutchinson and Neale

(1985)[61]later extended Hutchinson‟s 1974 work to

the conditions needed for the onset of wrinkling in

doubly curved sheet metal without lateral constraint.

They found that wrinkling is most likely a local

instability problem depending on the local curvature

and local stress states this finding applies to plates

and shells whose top and bottom surfaces are free of

contact. The problem differs slightly when we

consider the effect of a lateral constraint (constraint

normal to the plane of the sheet) on buckling in the

case of sheet metal forming, lateral constraints are

present in the form of binder/sheet/die or

die/sheet/die interactions where the sheet is

constrained to some extent between a binder and a

die or matching dies via force controlled or

displacement controlled condition. Our previous

experimental and numerical work (Jalkh et l.,

1993[62]) showed that wrinkling behavior depends

upon the pressure applied normal to the sheet by the

binder, as well as the local stress states and curvature.

The closest analytical solution for this situation is

given by Triantafyllidies and Needleman(1980)[63].

They applied the bifurcation theory to an annular

plate subjected to axisymmetric radial tension along

its inner edge. By resting the annular plate on a

continuous linear elastic foundation, they treated the

binder as a deformable binder and obtained the effect

of binder stiffness on a critical buckling stress and the

wave number. Also they assume small strain

deformation. Unfortunately, n direct comparison

between the analytical results and experimental

results were given.

3.4 Blank holding force

By understanding the mechanics of metal

deformation in sheet metal forming and having

condense in numerical simulations, the next logical

step is to design the Process using numerical

simulations. Very early attempts utilized the slip line

theory [64], upper bound analysis, and lower bound

analysis. An inverse method using deformation

theory was developed for designing the initial blank

shape in [65]. Based on the minimum plastic work

principle, Chung and Richmond [66] proposed an

ideal forming theory to design the optimal blank

shape and forming procedures [67]. With the

development of data acquisition systems and

computer hardware, the concept of variable binder

force was introduced by Hirose et al. [68]. More

recently, the ability to change the binder force during

the forming process is no longer limited to a research

apparatus stamping plants. However, most of the

work found in the literature utilized a force trajectory

generated by a trial-and-error approach. Using a

control element in their FEM model, Cao and Boyce

[69] monitored the tendency of wrinkling and tearing




1985 | P a g e

and thereby, designed a single variable binder force

trajectory for a conical cup forming using numerical

simulation. This trajectory was implemented in

experiments and led to a 16% increase in the

Ultimate forming height of the cup over the

traditional processes.

Fenn and Hardt [70] developed a real-time

closed-loop control system to alter the binder force

during the forming process using the actual punch

force or material draw-in as inputs. They obtained

consistent forming heights despite the presence of

variations in the lubricant, blank location and initial

binder force. This approach was adopted successfully

by Jalkh et al. [71] for aluminum cup forming.

Spring back can be reduced through

modifications to the forming process. Several

researchers have proposed to use a stepped binder

force trajectory to accomplish this objective

(Ayres[72], 1984; Hishida and Wagoner[73], 1993;

Sunseri[74] et al., 1996). A stepped binder force

trajectory is an instantaneous jump from a low binder

force (LBF) value to a high binder force (HBF) at a

specified percentage of the total punch displacement

(PD%) . Sunseri et al. (1996) but is also available in

commercial stamping presses, which can be found in

all the major automotive companies' investigated

springback in the Aluminum channel forming

process. Their work was conducted through

experiments and simulations at specific values for

process and material parameters.

For the constant binder force (CBF) cases

there exists an optimal CBF which results in a max

cup height with out wrinkling or tearing failure.

However, CBF experiments do not necessarily

optimize material flow trajectories for forming

without failure. Hirose[75] et al. (1990) examined

binder force trajectories containing step changes

while forming full scale automobile panels .All

trajectories that did succeed were those that started at

low binder forces and then increased during the later

stage of the process. The results obtained were

successful but were obtained by intensive trail and

error. Much recent work on sheet metal has focused

on designing variable binder force (VBF) histories

with majority papers reports a success in

experimental trail and error approach.

Another promising method to determine the optimal

binder force trajectory in the forward mode was

presented in Cao and Boyce [76] for a given initial

blank geometry and tooling geometry. A user-

defined element serves the purpose of closed-loop

control in the FEM simulation. This Element senses

the current wrinkling and tearing tendencies and

alters the binder force so that no Excessive wrinkles

are present. The optimal binder force history was

determined in one run of the FEM simulation.

IV. MATHEMATICAL FORMULATION

In the present formulation the

BELYTISCHKO Lin Stay shell element from LS

DYNA IS USED The element has 5 through the

thickness integration points and it requires 725

mathematical operations.

BELYTISCHKO shell element is based on a

combined rotational and strain formulation the

efficiency of the element is obtained from the

mathematical simplifications that result from these

two kinematical assumptions. The co rotational

portion of the formulation avoids the complexity of

non linear mechanics by embedding a coordinate

system in the element. The choice of velocity strain,

or rate of deformation, in the formulation facilities

the constitutive evaluations, since the conjugate stress

is more familiar Cauchy stress.

4.1 Co rotational coordinates

The mid surface of quadrilateral shell

element , or reference surface, is defined by the

location of elements four corner nodes . An

embedded element coordinate system as shown in

figure that deforms with the element is defined in

terms of this local coordinates. Then the procedure

for constructing the co rotationalco ordinate system

begins by caliculating a unit vector nrmal to the main

diagonal of the element

ê 3 = 3

3

s

s

(4.1a)

3s = 2

33

2

32

2

31 sss (4.1b)

42313 Xrrs (4.1c)

Where the super script caret(

. ) is used to indicatethe

local(element) coordinate system.

It is desired to establish the local x axis

x

approximately along the element edge between nodes

1 and 2. This definiton is convenient for interpreting

the element stresses which are defined in local

yx coordinate system.the procedure for

constructing this unit vector is to define a vector

1s that is nearly parallel to the vector 21r .

s .( 21211 rr ê 3 )ê 3 (4.2a)

ê

11

11

s

s (4.2b)




1986 | P a g e

Fig 4.1 construction of element coordinate system

The remaining unit vector is obtained from the vector

cross product.

ê 2 =ê 3 X ê 1 (4.3)

If the four nodes of the element are coplanar, then the

unit vectors 1e and 2e are tangent to the midplane of

the shell and 3

e is in the fiber direction. As the

element deforms, an angle may develop between the

actual fiber direction and the unit normal 3

e . The

magnitude of this angle may be characterized as

ê 3 . 1f <δ (4 .4)

Where f is the unit vector in the fiber direction and

the magnitude of depends on the magnitude of the

strains. According to Belytschko et al, for most

engineering applications acceptable values of are

on the order of 210

and if the condition presented

in Equation (4.4) is met, then the difference between

the rotation of the co-rotational coordinates

e and

the material rotation should be small. The global

components of this co-rotational triad define a

transformation matrix between the global and local

element coordinate systems. This transformation

operates on vectors with global components A=

( zyx AAA ) and element coordinate

components A=( zyx AAA ) and is defined as;

A =

z

y

x

A

A

A

=

zzz

yyy

xxx

eee

eee

eee

321

321

321

z

y

x

A

A

A

=

AqAT

(4.5a)

Where iZiYiX eee ,, the global components

of the element are coordinate unit vectors . The

inverse transformation is defined by the matrix

transpose, i.e.

AAT

(4.5b)

4.2 Velocity-Strain Displacement Relations

The above small rotation condition s,

Equation (4.4), does not restrict the magnitude of the

element‟s rigid body rotations. Rather the restriction

is placed on the out-of-plane deformations, and thus

on the element strain. Consistent with this restriction

on the magnitude of the strains, the velocity-strain

displacement relations used in the Belytschko-Lin-

Stay shall are also restricted to small strains.

As in the Hughes –Liu shell element, the

displacement of any point in th4 shell is partitioned

into a mid surface displacement (nodal translations)

and a displacement associated with rotations of the

element‟s fibers (nodal rotations). The Belytschko-

Lin-Tsay shall element uses the Mindlin (1951)

theory of plates and shells to pertition the velocity of

any point in the shell as

3ezm

X (4.6)

Where mV is the velocity of themed-surface, is

the angular velocity vector, and

z is the distance

along the fiber direction (thickness ) of the shell

element. The corresponding co-rotational

components of the velocity strain (rate of

deformation) are given by

i

j

j

i

ij

xx

d

2

1 (4.7)

Substitution of Equation (3.6) into the above yields

the following velocity-strain relations.

x

dd

x

m

x +

x

zy

(4.8a)

y

z

y

dx

m

y

x

(4.8b)

xy

z

xy

dxy

m

ym

xxy

2 (4.8c)

xz

m

xy

y

d

2 (4.8d)

yz

m

xz

x

d

2 (4.8e)

The above velocity-strain relations need to

be evaluated at the quadrature points within the Shell.




1987 | P a g e

Standard bilinear nodal interpolation is used to define

the mid surface velocity, angular Velocity, and the

element‟s coordinates (isoperimetric representation).

These interpolations relations are given by

II

m VNV , (4.9a)

II

m N , (4.9b)

II

m XNX , (4.9c)

Where the subscript I is summed over all the

element‟s nodes and the nodal velocities are obtained

by differentiating the nodal coordinates with respect

to time, i.e. II x.

. The bilinear shape functions

are4

)1)(1(4

11 N (4.10a)

(4.10 b)

)1)(1(4

13 N (4.10c)

)1)(1(4

14 N (4.10d)

The velocity strains at the center of the

element, i.e., at 0 and 0 are

obtained by substitution of the above relations into

the previously defined velocity-strain displacement

relations, Equations (4.8a) and (4.8e). After some

algebra, this yields

yIIxIIx BzBd

11 (4.11a)

xIIyIIy BzBd

22 (4.11b)

xIIyIIyIIxIIxy BBzBBd 12122

(4.11c)

yIIzIIxz NBd

12 (4.11d)

yIIzIIyz NBd

22 (4.11e)

Where,

x

NB I

11 (4.12a)

y

NB I

12 (4.12b)

The shape function derivative aIB are

also evaluated at the center of the element i.e. at

0 and 0

3.3. Stress Resultants and Nodal Forces. After suitable constitutive evaluations using

the above velocity strains, the resulting stresses are

integrated through the thickness of the shell to obtain

local resultant forces and moments. The integration

formula for the resultants is

zdfR

(4.13a)

zdzm

R

(4.13b)

Where the superscript R indicates a resultant

force or moment and the Greek subscripts emphasize

the limited range of the indices for plane stress

plasticity.

The above element –centered force and

moment resultants are related to the local nodal

forces and moments by invoking the principle of

virtual power and performing a one-point quadrature.

The relations obtained in this manner are

R

xyI

R

xxIx fBfBAf 211 (4.14 a)

R

xyI

R

yyIy fBfBAf 121 (4.14 b)

R

yzI

R

xzIzfBfBAkf 211

(4.14 c)

R

yz

R

xyI

R

yyIxI fk

mBmBAm4

12 (4.14d)

R

xz

R

xyI

R

xxIyI fk

mBmBAm4

21 (4.14 e)

0

zIm (4.14 f)

Where A is the area of the element and k is

the shear factor from the Mindlin theory. In the

Belytschko-Lin –T say formulation, k is used as a

penalty parameter tol enforce the Kirchhoff normality

condition as the shell becomes thin.

The above local nodal forces and moments

are then traansformed to the global coordinate system

usiing the transformation relations given previously

as Equation (4.5a) . The globala nodal forces and

moments are then appropriately summed over all the

nodes and the global equation of motion are solved

for the next increment in nodal accelerations.

In the present formulation 2 material properties has

been used

1. Power Law Isotropic Plasticity

2. Rigid

)1)(1(4

12 N




1988 | P a g e

1. Power Law Isotropic Plasticity:

Elastoplastic behavior with isotropic

hardening is provided by this model. The yield

stress, Y , is a function of plastic strain and obeys

the equation:

np

yp

n

y

Where yp is the elastic strain to yield and p

is

the effective plastic strain (logarithmic).

A parameter, SIGY, in the input governs

how the strain to yield is identified. If SIGY is set to

zero, the strain to yield if found by solving for the

intersection of the linearly elastic loading equation

with the strain hardening equation:

Where SIGY yield stress

E n

This gives the elastic strain at yield as:

1

1

n

yp

E

If SIGY yield is non zero and greater than 0.02 then:

ny

yp

1

Strain rate is accounted for using the

Cowper and Symonds model which scale the yield

stress with the factor

1+p

C

1

Where is the strain rate. A fully viscoplastic

formulation is optional with this model which

incorporates the Cowper and Symonds formulation

within the yield surface. An additional cost is

incurred but the improvement is results can be

dramatic.

2. Rigid:

The rigid material type provides a

convenient way of turning one or more parts

comprised of beams, shells, or soled elements into a

rigid body. Approximating a deformable body as

rigid is a preferred modeling technique in many real

world applications.

For example, in sheet metal forming

problems the tooling can properly and accurately be

treated as rigid. In the design of restrain systems the

occupant can, for the purposes of early design

studies, also be treated as rigid. Elements which are

rigid are bypassed in the element processing and no

storage is allocated for storing history variables;

consequently, the rigid material type is very cost

efficient.

Two unique rigid part ID‟s may not share

common nodes unless they are merged together using

the rigid body merge option. A rigid body may be

made up of disjoint finite element meshes, however.

LS-DYNA assumes this is the case since this is a

common practice in setting up tooling meshes in

forming problems.

All elements which reference a given part

ID corresponding to the rigid material should be

contiguous, but this is not a requirement. If two

disjoint groups of elements on opposite side of a

model are modeled as rigid, separate part ID‟s should

be created for each of the contiguous element groups

if each group is to move independently. This

requirement arises from the fact that LS-DYNA

internally computes the six rigid body degrees-of-

freedom for each rigid body (rigid material or set of

merged materials), and if disjoint groups of rigid

elements use the same part ID, the disjoint groups

will move together as one rigid body.

Inertial properties for rigid materials may be

defined in either of two ways. By default, the inertial

properties are calculated from the geometry of the

constituent elements of the rigid material and the

density specified for the part ID. Alternatively, the

inertial properties and initial velocities for a rigid

body may be directly defined, and this overrides data

calculated from the material property definition and

nodal initial velocity definitions.

Young‟s modulus, E, and Poisson‟s ratio, V,

are used for determining sliding interface parameters

if the rigid body interacts in a contact definition.

Realistic values for these constants should be defined

since unrealistic values may contribute to numerical

problem in contact.

V. DESIGN CALCULATIONS The material properties assigned to blank are

shown in table 5.1 and the material properties

assigned to remaining are considered as rigid and the

schematic view of the model is shown in fig 5.1

Table 5.1: Material properties assigned to blank




1989 | P a g e

Fig 5.1: Schematic view

The following factors may be calculated

with formulas presented here:

5.1 Blank diameter

a. Prepared formula

b. Method for intricate cups

5.2 Percent reduction

a. Drawing

b. Redrawing

5.3 drawing force

a. Drawing force

b. Blank holding force

c. Ironing force.

The formulas presented are applicable to the

cupping operation only. They may be adjusted, in

some cases to approximate forces for irregular shaped

panels.

5.1 Blank diameter

The first step in preparing for a cupping

operation is find of blank diameter required. If a

finished cup is available, it could weight. The blank

then has the same weight as the cup. A blank of

correct thickness would be cut to the diameter

necessary to produce this weight.

Several prepared formulas are available for

calculating the blank diameter required for various

shapes of cups, as follows:

D = blank diameter

d = punch diameter

h = cup height

D = = 67.08mm

5.2 Allowances for Trimming

As described in the theory of drawing,

additional metal must be providing for trimming.

Excess metal is needed for gripping. The metal

becomes scored and scratched and must be removed.

Therefore, after the blank diameter required for the

finished cup has been found, the diameter is

increased an amount sufficient for trimming

operation. This is called trim allowance.

The larger the cup being drawn, the larger

the trim allowances should be. The prediction of

blank size for larger cups may be less certain.

Approximate trim allowances are as fallows.

Table 5.1: Approximate trim allowances

The trim allowance varies with the metal being

drawn. Since the trim allowance used is directly

related to the scrap percentage produced, care must

be taken in setting this allowance.

The final blank diameter is then as fallows:

Blank diameter =Development blank diameter +

Trim allowance

5.3 Percentage reduction:

When the final blank diameter is known, the

next step is to determine the percentage of reduction

necessary to convert the blank into the desired cup. If

the allowed percent reduction for drawing does not

produce the final desired cup size, then redraws must

be made. The allowable percent of reduction for the

redraw determine how many redraws are required. As

described in the theory of drawing operation, the

severity of cupping is determined by the relationship

between the punch steel diameter and blank diameter.

The closer these diameters are, the less severe is the

operation.

The percentage reduction is an expression of this

severity, as fallows:

D-d

% Reduction = x 100= 55.27

D

5.4 Thickness to diameter ratio:

Another factor controlling the severity of

drawing the ratio of the blank thickness to the blank

diameter. When two blanks of equal diameter are to

be drawn, the thicker blank can have a higher percent

reduction without wrinkling or cup failure. The blank

thickness is found as a percent of the blank diameter,

as follows:

t = sheet metal thickness

D = blank diameter

49.1100D

t

5.5 Drawing force:




1990 | P a g e

During drawing operation, the side wall is

placed tension to cause bending at the radii,

overcome friction and compress the metal in the

flange. Therefore, the logical way to calculate the

drawing force would be to use the tensile strength of

the metal.

The force needed to draw a cup is equal to

the product of the cross sectional area and the yield

strength is in tension of the work material. For a

cylindrical cup, the drawing force can be determined

by the formula

F =∏ d t σy (D/d – c) Newtons

Where, c= drawing force constant (varies from 0.52

to 0.7)

F = ∏x30 x1x 242 (67.08/30 – 0.7)

F = 35015.27 N

F = 35.015KN

Therefore Blank holding force =F/3 =11.671KN

Clearance: Clearance is usually 7 to 14% greater than

the sheet thickness i.e. 0.01mm.

VI. FINITE ELEMENT SIMULATIONS A finite element explicit solver LS Dyna is

used for the simulation of effect of various

parameters on the optimum blank holding force. In

this work the effect of friction coefficient, die radius,

clearance, and sheet thickness is studied. The

following methodology is applied for the same.

Tolling design is carried out for the cup of 30mm

diameter and 30 mm deep.

Blank holding force is calculated empirically.

Modeling is done in CATIA, imported the same

to VPG 3.1, a preprocessor for LS DYNA solver.

In the figures 6(a) we can see the wrinkles

formed at different blank holding force but at

same coefficient friction. and in fig 6(b) we can

see that there are no wrinkles on the flange of

cup this cup is taken at calculated blank holding

force i.e. 11671

The finite element model with different mesh

sizes are shown in fig 6(c).and the wire frame

model is shown in fig 6(d).

Punch, die and blank holder are selected as rigid

materials and material model selected for the

blank is as power law plasticity model ( σ = K*

Єn) with the properties shown in table 6.1

All elements are selected as shell elements. The

meshing parameters of the die, punch and blank

holder are shown in table 6.2

Optimum mesh size is found out by considering

stability of one of the out put parameters (max

von-mise‟s stress) and the selected mesh size is

used for the analysis.

Each parameter is varied keeping other three

parameters constant (incase of thickness, instead of

clearance, % of clearance is taken as constant) the

results are presented and compared.

Fig 6(a) Cup with wrinkles at 9500 BHF

Fig 6(b) Cup without wrinkles at 11671 BHF




1991 | P a g e

Fig 6(c): Finite element model

Fig 6(d): Wire frame model

The fig 6(a) is taken at 9500 BHF with 0.15

coefficient friction at 5th

time step .and fig 6(b) is

taken with calculated BHF (11671) at 0.15

coefficient friction and at same time step i.e.5th

those

are shown at top left corner of those figures. In fig

6(a) we can se wrinkles where as in case of fig6 (b)

there are no formation of wrinkles.

video 1.avi

video 2.avi

video 3.avi

6.1. Selection of optimum mesh size for the blank

Since a numerical solution is an approximate

one which may be converged to its true value by

refining the mesh or by using higher order elements.

In this work the convergence is checked by refining

the mesh. It is carried out by studying the stability of

one of the out put parameters. The out put parameter

chosen here is Max von-mise‟s stress. Mesh chosen is

topology mesh with the max size of 3mm initially.

The simulation is carried out and the max vonmise‟s

stress is noted at each step and plotted in fig 6.1. It is

found that the fluctuation is high. Then the mesh size

is reduced successively in steps of 0.25 mm and the

corresponding values against the punch stroke are

plotted (fig 6.1)

Table 6.1: Meshing parameter

Fig 6.1: Max vonmise‟s stress v/s punch stroke for

various mesh sizes

From the fig 6.1 it is observed that the

variation of the max von mise‟s stress is almost stable

for the mesh size of 2mm with the parameters shown

in table 6.3

6.2 Selecting optimum blank holding force at

different coefficient of friction: In this case

checking of wrinkles has been done at




1992 | P a g e

different blank holding force with different

coefficient of friction. The blank holding

force in table 6.2 is the one where there is

no formation of wrinkling at different

coefficient of friction.

Table 6.2: Optimum Blank Holding Force Vs

Coefficient Of Friction

Fig 6.2: variation of blank Holding Force with

coefficient of friction

From fig 6.2 it is observed that the Optimum

B H F decreases with increase of coefficient of

friction. It is found that up to the friction co efficient

of 0.05the decrease is less from 0.06 to 0.13 there is

linear variation .Optimum B H F is inversely

proportional to coefficient of friction. Either decrease

is linear beyond that the optimum B H F decreases

drastically.

6.3 Selecting optimum blank holding force at

different die radius

From fig 6.3 it is evident that the optimum B H F

decreases drastically with increase of die radius at

smaller die radius. After that it will not affect

optimum B H F.




1993 | P a g e

6.4 Variations of vonmises stresses with blank

holding force at different coefficient of friction:

Table 6.4: Variations of vonmises stresses with

blank holding force at Different coefficient of

friction.

C

Fig 6.4: variation of vonmises stresses with Blank

Holding Force at Different Coefficient of friction

From fig (6.4) it is observed that the max

vonmises stress is occurring at the die corner. From

the fig it is known that the B H F has no effect on the

max vonmises stresses. Since its max value is

observed at the die corner. Same is evident for

maximum plastic strain in fig (6.5). The vonmises

stress and plastic strain may be compared at side

walls from fig. (6.6) to it is evident that max plastic

strain decreases with the die radius and increases

further. So there is an optimum die radius where

plastic strain for a given BHF is minimum. The same

trend is observed in max vonmises stress (fig. 6.7).

Table 6.5: variation of plastic strain with Blank

Holding Force at different Coefficient of friction

Fig 6.5: variation of plastic strain with Blank

Holding Force at Different Coefficient of friction.




1994 | P a g e

Table 6.6 variation of vonmises stresses with Blank

Holding Force at Different Die radius

Fig 6.6: variation of vonmises stresses with Blank

Holding Force at Different die radius

Table 6.7: variation of plastic strain with Blank

Holding Force at Die radius

Fig 6.7: variation of plastic strain with Blank Holding

Force at Die radius

VII. CONCLUSION The conclusion of thesis work are

enumerated and presented as shown.




1995 | P a g e

For a given set of punch die and working

conditions there exists an optimum blank holding

force which prevents the wrinkles and at same

from the stresses induced in the cup is minimum.

Blank holding force decreases with increase of

coefficient of friction for a small range of

coefficient of friction and a linear relation exists.

With the increase of die radius up to a certain

value Optimum BHF drastically reduces beyond

that it remains constant.

There will be an optimum die radius where max

plastic strain and max vonmises are minimum.

REFERENCES [1] Langerak, 1999a .

[2] Haar R. ter, „Friction in sheet metal

forming: the influence of (local) contact

conditions and deformation‟, Ph.D. Thesis,

University of Twente, Enschede, ISBN 90-

9009296X, 1996.

[3] Vreede, 1992] Vreede P.T., „A finite

element method for simulations of 3-

dimensional sheet metal forming‟, Ph.D.

Thesis, University of Twente, Enschede,

ISBN 90-90047549, 1992 ).

[4] Vegter H., Y. An, H.H. Pijlman, J. Huétink,

„Advanced mechanical testing on

aluminium alloys and low carbon steels for

sheet forming‟, Proceedings of the 4th

International Conference on Numerical

Simulations of 3-D Sheet Metal Forming

Processes, J.C. Gelin & P. Picard (eds.),

Besançon, vol. 1, p. 3-8, 1999.

[5] Wang N-M, Budiansky B. Analysis of sheet

metal stamp;ing by a finite element method.

Journal of Applied Mechanics, ASME

Transactions 1978;45:73-82.

[6] Budiansky B. Notes on nonlinear shell

theory. Journal of Applied Mechanics

1968;35:393-401.

[7] Hughes TJR, Liu WK. Nonlinear finite

element analysis of shells: Part I. Three

dimension al shells. Computer Methods in

Applied Mechanics and Engineering 1981;

26:331-62.

[8] Belytschko T.Stolarski H.Carpenter N.A C

triangular plate element with one-point

quadrature. International Journal for

Numerical Methods in Engineering

1984;20:787-802.

[9] Onate E. Zienkiewicz OC A viscous

formulation for the analysis of thin sheet

metal forming. Internations Journal of

Mechanical Sciences 1983;25(5):305-35.

[10] Stoughton T M. Finite Element Modeling of

1008 AK Steel stretched over a rectangular

punch with bending effects In: Wang NM,

Tang SC , editors. Computer modeling of

sheet metal forming processes; theory

verification and application. Proceedings of

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